Subset

Definition

A set $A$ is a subset of a set $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$.

\[A \subseteq B \iff \forall x \, (x \in A \implies x \in B)\]

For any set $A$: $$A \subseteq A$$

If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

If $A \subseteq B$ and $B \subseteq A$, then $A = B$.

$A$ is a proper subset of $B$, written $A \subsetneq B$, if $A \subseteq B$ and $A \neq B$.

\[\{1, 2\} \subseteq \{1, 2, 3\}\]

$$\{1, 2, 3\} \subseteq \{1, 2, 3\}$$ (but not proper)

$$\emptyset \subseteq A$$ for any set $A$.

For every set $A$, $\emptyset \subseteq A$.

We must show: $\forall x \, (x \in \emptyset \implies x \in A)$.

Since $x \in \emptyset$ is always false, the implication is vacuously true.

QED